Because of difficulties with the Gupta-Bleuler subsidiary condition in the charged sectors, an alternative scheme for identifying physical states in the indefinite-metric space I of quantum electrodynamics is proposed: Any vector ΦI is a physical state if it is positive on the observables, θΦ,θΦ0, Φ,Φ=1, for θ any element of the algebra of observables. Observables θ, in turn, are selected by the requirement that they commute with the generators of the restricted gauge transformations of the second kind, Aμ→Aμ+μλ, ψ→ψexp(ieλ), with λ(x)=c-number, 2λ=0. This is equivalent to the requirement [B(x),θ]=0, where B(x)=•A(x) in the Feynman gauge. It is proved that the substitute Gupta-Bleuler condition B(-)(x)Φ=b(-)(x)Φ provides a subspace I[b] of physical states, where b(-)(x) is the negative-frequency part of any real c-number solution of the wave equation 2b(x)=0 satisfying b(x)d3x=q, with q an eigenvalue of the charge operator. Different functions b(x) characterize different superselection sectors which are eigenspaces of generators G(λ) of the restricted gauge transformations of the second kind with eigenvalues G(λ)=λ(x)0b(x)d3x. In a given superselection sector Maxwell's equations take the form μFμν=Jν-νb, where -νb is interpreted as a classical external current which is induced by the quantum-mechanical current Jν. The proof relies on the axiom of asymptotic completeness I=Iin=IoutandIin and is specified by the ansatz of infrared coherence, namely, limω→0aμin(k)∼-(2π)-32ieipipi•k, where aμin(k) is the photon annihilation operator and pi is the momentum of an incoming particle of charge ei, and in → out. The spectral decomposition of the infrared-coherent space is effected. Its singularity in the neighborhood of the electron mass agrees with the singularity of the electron propagator in the Feynman gauge, which allows an on-shell normalization of the charged field ψ.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)